Question

Consider two conducting spheres of radii $R_1$ and $R_2$ with$R_1>R_2$. If the two are at the same potential, and the larger sphere has more charge than the smaller sphere, then

• A. the charge density of smaller sphere is less than that of larger sphere
• B. the charge density of smaller sphere is more than that of larger sphere
• C. both spheres may have same charge density
• D. none of these

Answer 1

Answer: (B)

Here, $V_1=V_2$ or $\frac{q_1}{4\pi {\epsilon }_0{R}_1}=\frac{q_2}{4\pi {\epsilon }_0{R}_2}$
$\therefore \frac{q_1}{q_2}=\frac{R_1}{R_2}$
Given $R_1>R_2$
$\therefore q_1>q_2$
$\therefore$ Larger sphere has more charge than the smaller sphere.Now charge densities
${\sigma }_1=\frac{q_1}{4\pi R_1^2}$and ${\sigma }_2=\frac{q_2}{4\pi R_2^2}$
$\therefore \frac{{\sigma }_2}{{\sigma }_1}=\frac{q_2}{q_1}\frac{R_1^2}{R_2^2}$
or $\frac{{\sigma }_2}{{\sigma }_1}=\frac{R_2}{R_1}\frac{R_1^2}{R_2^2}=\frac{R_1}{R_2}$ (using $\left(i\right)$)
As $R_1>R_2$ therefore ${\sigma }_2>{\sigma }_1$
Charge density of smaller sphere is more than the charge density of larger sphere